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Integrable structure of conformal field theory, quantum KdV theory and Thermodynamic Bethe Ansatz

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We construct the quantum versions of the monodromy matrices of KdV theory. The traces of these quantum monodromy matrices, which will be called as “T-operators,” act in highest weight Virasoro modules. TheT-operators depend on the spectral parameter λ and their expansion around λ=∞ generates an infinite set of commuting Hamiltonians of the quantum KdV system. TheT-operators can be viewed as the continuous field theory versions of the commuting transfermatrices of integrable lattice theory. In particular, we show that for the values\(c = 1 - 3\frac{{3(2n + 1)^2 }}{{2n + 3}}\),n=1,2,3 .... of the Virasoro central charge the eigenvalues of theT-operators satisfy a closed system of functional equations sufficient for determining the spectrum. For the ground-state eigenvalue these functional equations are equivalent to those of the massless Thermodynamic Bethe Ansatz for the minimal conformal field theoryM 2,2n+3; in general they provide a way to generalize the technique of the Thermodynamic Bethe Ansatz to the excited states. We discuss a generalization of our approach to the cases of massive field theories obtained by perturbing these Conformal Field Theories with the operator Φ1,3. The relation of theseT-operators to the boundary states is also briefly described.

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Author notes

  1. Vladimir V. Bazhanov

    Present address: Institute for High Energy Physics, 142284, Protvino, Moscow Region, Russia

Authors and Affiliations

  1. Department of Theoretical Physics and Center of Mathematics and its Applications, IAS Australian National University, 0200, Canberra, ACT, Australia

    Vladimir V. Bazhanov

  2. Newman Laboratory, Cornell University, 14853-5001, Ithaca, NY, USA

    Sergei L. Lukyanov

  3. Department of Physics and Astronomy, Rutgers University, 08855-049, Piscataway, NJ, USA

    Alexander B. Zamolodchikov

  4. L.D. Landau Institute for Theoretical Physics, 142432, Chernogolovka, Russia

    Alexander B. Zamolodchikov

Authors
  1. Vladimir V. Bazhanov
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  2. Sergei L. Lukyanov
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  3. Alexander B. Zamolodchikov
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Communicated by M. Jimbo

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Bazhanov, V.V., Lukyanov, S.L. & Zamolodchikov, A.B. Integrable structure of conformal field theory, quantum KdV theory and Thermodynamic Bethe Ansatz. Commun.Math. Phys. 177, 381–398 (1996). https://doi.org/10.1007/BF02101898

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